1
Collective Human Mobility Pattern from Taxi Trips in Urban Area Chengbin Peng1,2 , Xiaogang Jin2 , Ka-Chun Wong1 , Meixia Shi3 , Pietro Li`o4,∗ 1 Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Jeddah, Kingdom of Saudi Arabia 2 Institute of Artificial Intelligence, College of Computer Science, Zhejiang University, Hangzhou, China 3 College of Environmental and Resource Sciences, Zhejiang University, Hangzhou, China 4 Computer Laboratory, Cambridge University, Cambridge, United Kingdom ∗ E-mail:
[email protected]
Supporting Information S3 Moment Generation Function of α In this section we discuss α as a random variable following a normalized binomial distribution described by Eq. (10) and Eq. (11). Then the moment generating function of α is M (g) =⟨eg×α ⟩ =
(
pn ∑
e
α⟨T N ⟩=0
=
pn ∑ α⟨T N ⟩=0
g×α
) pn rα⟨T N ⟩ (1 − r)pn−α⟨T N ⟩ α⟨T N ⟩
(
) g pn (r × e ⟨T N ⟩ )α⟨T N ⟩ (1 − r)pn−α⟨T N ⟩ α⟨T N ⟩
(1)
g
=(r × e ⟨T N ⟩ + (1 − r))pn g
=(r × e pn×r + (1 − r))pn and consequently, the first, second and third moments are as follows. µ1 =M ′ (g)|g=0 =(r + (1 − r))pn−1 =1
(2)
µ2 =M ′′ (g)|g=0 1 1 + =(pn − 1) pn ⟨T N ⟩ 1 ≈ ⟨T N ⟩
(3)
2
µ3 =M ′′′ (g)|g=0 pn − 1 1 2 = [(pn − 2)r + ] pn ⟨T N ⟩ ⟨T N ⟩ 1 1 1 [(pn − 1)r + ] + ⟨T N ⟩ ⟨T N ⟩ ⟨T N ⟩ 2 1 1 ≈1 + + (1 + ) ⟨T N ⟩ ⟨T N ⟩ ⟨T N ⟩ 3 1 =1 + + ⟨T N ⟩ (⟨T N ⟩)2 Typically pn is quite large, so that pn ≈ pn − 1 ≈ pn − 2, and the approximations above are valid.
References
(4)
